ASPECTS OF TEICHMUELLER THEORY

April 12-14, 2007
University of Michigan
Ann Arbor, MI

       

 

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University of Michigan

Mathematics Department


Organizers
Dick Canary
Zeno Huang
Lizhen Ji
Ralf Spatzier


 

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Schedule
Thursday, April 12 - B844 East Hall
  4:10 - 5 PM   Wolpert  

The Weil Petersson Geometry for Teichmueller Space
An overview of the WP local geometry for the space of marked Riemann surfaces will be presented. Curvature properties of the metric and convexity properties of geodesic-length functions will be considered. The CAT(0) geometry of the completion, the augmented Teichmueller space, will be described. The augmented space is the closed convex hull of the maximal points in the bordification. A proof of the Masur-Wolf theorem on WP isometries will be presented.

           
Friday, April 13 - B844 East Hall
  2:00 - 3:00 PM   Wolf   Local and Global Computations in Teichmuller Space.
We discuss complex analytic and Riemannian approaches to computing the changes of invariants on Riemann surfaces as the conformalor hyperbolic structure changes. We apply these methods in a few examples.
  3:15 - 4:15 PM   Luo  

New coordinates of Teichmuller Spaces of Surfaces with Boundary
In our recent work, we introduced, for each real
number t, coordinate C_t of the Teichmuller space of a surface
with boundary. These coordinates arise naturally from
variational principles associated to a surface with ideal
triangulation. It is proved that, for non-negative t, the images of the Teichmuller space in these coordinates C_t are a open convex polytope independent of t. Many interesting questions about these coordinates will be discussed.

  4:30 - 5:30 PM   Wolpert  

A Medley of WP Formulas
Formulas will be presented for the pairing of geodesic-length gradients, for geodesic-length Hessians and for the WP connection. Comparability models for the metric will be presented. Properties of geodesics terminating at the augmentation locus will be discussed. The Alexandrov tangent cone of the augmented Teichmueller space will be described. A fixed point theorem for group actions will be described.

           
Saturday, April 14 - B844 East Hall
  9:30-10:30 AM  

Luo

 

  Grothendieck's Reconstruction Principle for Teichmuller Spaces
According to Grothendieck in his Esquisse d'un programme, surface theory should be built on two simple surfaces, i.e., the 1-holed torus and the 4-holed sphere. We will try to explain Grothendieck's idea using a result on Teichmuller
spaces.
  11:00 AM-
12:00 PM
  Wolf  

Degeneration in Teichmuller Space and Applications.
We describe some compactifications to spaces of Riemann
surfaces, with some applications outside Riemann surface theory.

           
  1:00 - 2:00 PM   Wolpert  

Estimating hyperbolic Green’s functions for degenerating surfaces (pdf)

  2:15 - 3:15 PM   Ivanov  

Teichmüller Spaces and Teichmüller Modular Groups I
The Teichmüller modular groups, also known as the
mapping class groups of surfaces, are the groups of the
isotopy class of self-diffeomorphisms of a surface. At the
same time they may be considered as the orbifold fundamental
groups of the moduli spaces of Riemann surfaces. The moduli
spaces are the quotients of Teichmüller spaces by the natural
action of Teichmüller modular groups.

We will describe how these actions can be used in order to understand the algebraic structure of Teichmüller modular groups (for example, to compute their cohomological dimension). A key ingredient is the complex of curves, which describes the combinatorics of degenerations of Riemann surfaces.

For the most of the time, we will approach Teichmüller spaces from the elementary viewpoint of the hyperbolic surfaces (as opposed to the complex-analytic point of view).

We will also make a connection with a topic discussed in the lectures of S. Wolpert, namely with the classical result of Royden which inspired the Masur-Wolf theorem about the WP isometries.

  3:30 - 4:30 PM  

Ivanov

 

  Teichmüller Spaces and Teichmüller Modular Groups II
(continuation)
           

 

Sponsors
Sponsored by the National Science Foundation via our RTG grant in Geometry, Topology and Dynamics, and the FRG Geometric Function Theory.
Also supported by the Department of Mathematics at the University of Michigan.

 

     

 


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Ann Arbor, MI 48109-1043
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